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Let me give you some context first: just a few days ago I found some intriguing references to Ricci flows in the setting of directed graphs.

There are at least two versions of Ricci curvature in the discrete realm (one being the Ollivier-Ricci curvature, the other the Forman-Ricci, see here for reference (*)), and as it turns out, they are both useful in graph analytics.

To be a tad more specific, one application leads to a new method for determining communities (the so-called Ricci communities, for the interested ones there is even a github Python implementation which can easily be used for hands-on explorations ), whereas another quite useful one is used to get rid of "bottlenecks" in graph messaging ( thereby solving some critical issue in Graph deep learning see picture below).

ricci flow on graph

https://towardsdatascience.com/over-squashing-bottlenecks-and-graph-ricci-curvature-c238b7169e16

Now, if I understand them correctly, the associated Ricci flow, just like in the differentiable realm, acts as a kind of "curvature heat-like operator", a diffusion which tends to smoothen out the curvature across the underlying geometrical object.

Perhaps naively, it occurred to me this:

why confining ourselves to diffusion? (note: I am aware of the centrality of the Ricci flow in the proof of the Poincare conjecture)

Could one replace the Ricci flow with some kind of PDE (or a difference equation in the finite setting) for the curvature change modeled on completely different PDEs?

For instance, what about a kind of wave equation?

Now the questions (and I apologize if this is too naive, I am coming from the data science world, my knowledge of Riemannian geometry does not go beyond standard grad courses):

  1. Have such curvature flow involving non-heat-like PDEs been investigated in the world of Riemannian geometry? I would think the answer is in the affirmative, but I just do not happen to know it.
  2. Are there any references for generalized curvature flows in discrete metric spaces and particularly in weighted directed graphs?

Any help is most welcome.

(*) actually in the referenced article there are three discrete Ricci curvatures, but I haven't wrapped my mind around the third one yet.

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    $\begingroup$ The point of heat flow is that, in an insulated body, we expect heat to settle rapidly into a final equilibrium state, which we imagine to be the state where the body is somehow in its most natural form. We hope that natural form to provide us with some geometric picture that understands the topology best, a best metric. But with a wave equation, or a Schrodinger equation, or a Yang-Mills equation, we expect it to just keep bouncing around. So there is no approaching a best metric. So the first question seems to be what we would hope for in a wave equation. $\endgroup$
    – Ben McKay
    Dec 5 '21 at 9:32
  • $\begingroup$ @BenMcKay thanks for your ultra-concise yet full of insights comment! I hear your point, loud and clear. Only thing is: best metric for what? Of course, Hamilton introduced the Ricci flow for proving the Poincare Conjecture, and Perelmann actually proved with it much more ( Geometrization of low-dim manifolds). So, I assume in that case the new metric is best in the sense that it reflects best the underlying topology. But I still feel there is more: other PDEs may reflect other insights, as suggested in the paper cited by Quarto $\endgroup$ Dec 5 '21 at 16:32
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    $\begingroup$ As for myself, as I have mentioned, my chief interest is the discrete case, weighted graphs. Now, imagine that the graph is a social network, or a financial one. I may be interested in modeling different things. Example: rather than looking for the best quasi-partitioning into static communities, I could look for their evolution. Here, sometimes you may find yourself in a strange scenario, in which communities are spread apart, and then, after a while, go back to a more cohesive status. I suspect that the curvature dynamics follows a wave-like pattern, rather than heat diffusion... $\endgroup$ Dec 5 '21 at 16:36
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    $\begingroup$ I would disagree with Ben McKay's comment a little bit, in that whenever Ricci flow produces something like a "best metric," you must be on a very special manifold and so from a certain perspective the "best metric" already trivially exists. But this point is more metaphysical than mathematical. It might be more useful to point out that all of Hamilton and Perelman's work on singularity formation is very important and useful and reveals deep structures, but does not actually produce best metrics, natural forms, equilibrium states etc., even in the form of the Thurston conjecture itself $\endgroup$ Dec 5 '21 at 17:14
  • $\begingroup$ @QuartoBendir thanks again for both your fantastic answer and your very last comment. Unfortunately I am like a drunkard walking into a new field: I see its beauty, but I am no master... Anyway Ben's comments has made me even more curious, especially on schrodinger flows.. Soon I will post another question along similar lines, and I sincerely hope that you two will come back. It is a great opportunity to be able to pick brains like yours, MO is indeed a great place to hang around... $\endgroup$ Dec 5 '21 at 17:46
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A small number of authors have considered hyperbolic versions of the standard flows, see e.g. "Wave character of metrics and hyperbolic geometric flow" by De-Xing Kong and Kefeng Liu and related articles.

I am not familiar with the literature on discrete curvature flows. Some papers that look interesting are "Super Ricci flows for weighted graphs" by Matthias Erbar & Eva Kopfer and "Simplicial Ricci flow" by Warner Miller, Jonathan McDonald, Paul Alsing, David Gu & Shing-Tung Yau. I would suspect that undirected graphs are more natural here than directed graphs, just based on analogy to Riemannian metrics.

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